Package lmesplines. R topics documented: February 20, Version

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1 Version Package lmesplines February 20, 2015 Title Add smoothing spline modelling capability to nlme. Author Rod Ball Maintainer Andrzej Galecki Description Add smoothing spline modelling capability to nlme. Fit smoothing spline terms in Gaussian linear and nonlinear mixed-effects models Depends nlme(>= ) License GPL (>= 2) Repository CRAN Date/Publication :08:38 Repository/R-Forge/Project nlmeu Repository/R-Forge/Revision 122 Repository/R-Forge/DateTimeStamp :37:01 NeedsCompilation no R topics documented: approx.z smspline smsplineex Index 7 approx.z Interpolating in smoothing spline Z-matrix columns Description Function to interpolate the Z-matrix for LME smoothing spline fits from one set of values of the time covariate to another. 1

2 2 approx.z Usage approx.z(z, oldtimes, newtimes) Arguments Z oldtimes newtimes Z-matrix with rows corresponding to the sorted unique values of the time covariate original (sorted) values for time covariate, corresponding to the rows of Z new (sorted) values for time covariate Details Uses linear interpolation of each column of the Z-matrix, regarded as a function of time, with the times given by oldtimes. Value A matrix corresponding to an interpolated spline matrix. Can be used with smspline() for fitting LME splines with random effects corresponding to a different set of values of the time covariate than those represented in the data, or as part of the newdata argument prediction from an existing model, to obtain predictions at points not represented in the data using predict.lme(). Note Linear interpolation works well here because the spline basis functions are approximately piecewise linear. Author(s) Rod Ball <rod.ball@scionresearch.com> See Also smspline nlme Examples times1 <- 1:10 Zt1 <- smspline(~ times1) times2 <- seq(1,10,by=0.1) Zt2 <- approx.z(zt1,oldtimes=times1,newtimes=times2)

3 smspline 3 smspline Smoothing splines in NLME Description Usage Functions to generate matrices for a smoothing spline covariance structure, to enable fitting smoothing spline terms in LME/NLME. Arguments smspline(formula, data) smspline.v(time) formula data time model formula with right hand side giving the spline covariate optional data frame spline time covariate to smooth over Details Value A smoothing spline can be represented as a mixed model (Speed 1991, Verbyla 1999). The generated Z-matrix from smspline() can be incorporated in the users s dataframe, then used in model formulae for LME random effects terms at any level of grouping (see examples). The spline random terms are fitted in LME using an identity pdmat structure of the form pdident(~z - 1). The model formulation for a spline in time (t) is as follows (Verbyla 1999): y = X s β s + Z s u s + e where X s = [1 t], Z s = Q(t(Q)Q) 1, and u s N(0, G s ), is a set of random effects. We transform the set of random effects u s to independence with u s = Lv s, where v s N(0, Iσ 2 s) is a set of independent random effects. The Z-matrix is transformed accordingly to Z = Z s L, where L is the lower triangle of the Choleski decomposition of G s. The function smspline.v() is called by smspline(), and can be used to access the matrices X s, Q, G s. See Verbyla (1999) for further information. For smspline(), a Z-matrix with the same number of rows as the data frame. After fitting, the LME model output gives a standard deviation parameter for the random effects, estimating σ s. The smoothing parameter from the penalised likelihood formulation is For smspline.v(), a list of the form λ = σ 2 /σ 2 s

4 4 smspline Xs Zs Q,Gs,R X-matrix for fixed effects part of the model Z-matrix for random effects part of the model Matrices Q, G s, R associated to the mixed-model form of the smoothing spline. Note The time points for the smoothing spline basis are, by default, the unique values of the time covariate. This is the easiest approach, and model predictions at the fitted data points, can be obtained using predict.lme. By interpolation, using approx.z, the Z-matrix can be obtained for any set of time points and can be used for fitting and/or prediction. (See examples). Synopsis:data$Z <- smspline(formula1, data); fit <-lme(formula2, data, random=... ) Author(s) Rod Ball <rod.ball@scionresearch.com> References The correspondence between penalized likelihood formulations of smoothing splines and mixed models was pointed out by Speed (1991). The formulation used here for the mixed smoothing spline matrices are given in Verbyla (1999). LME/NLME modelling is introduced in Pinheiro and Bates (2000). Pinheiro, J. and Bates, D. (2000) Mixed-Effects Models in S and S-PLUS Springer-Verlag, New York. Speed, T. (1991) Discussion of That BLUP is a good thing: the estimation of random effects by G. Robinson. Statist. Sci., 6, See Also Verbyla, A. (1999) Mixed Models for Practitioners, Biometrics SA, Adelaide. approx.z nlme Examples # smoothing spline curve fit data(smsplineex1) # variable all for top level grouping smsplineex1$all <- rep(1,nrow(smsplineex1)) # setup spline Z-matrix smsplineex1$zt <- smspline(~ time, data=smsplineex1) fit1s <- lme(y ~ time, data=smsplineex1, random=list(all=pdident(~zt - 1))) summary(fit1s) plot(smsplineex1$time,smsplineex1$y,pch="o",type="n", main="spline fits: lme(y ~ time, random=list(all=pdident(~zt-1)))", xlab="time",ylab="y") points(smsplineex1$time,smsplineex1$y,col=1) lines(smsplineex1$time, smsplineex1$y.true,col=1) lines(smsplineex1$time, fitted(fit1s),col=2)

5 smsplineex1 5 # fit model with cut down number of spline points times20 <- seq(1,100,length=20) Zt20 <- smspline(times20) smsplineex1$zt20 <- approx.z(zt20,times20,smsplineex1$time) fit1s20 <- lme(y ~ time, data=smsplineex1, random=list(all=pdident(~zt20-1))) # note: virtually identical df, loglik. anova(fit1s,fit1s20) summary(fit1s20) # model predictions on a finer grid times200 <- seq(1,100,by=0.5) pred.df <- data.frame(all=rep(1,length(times200)),time=times200) pred.df$zt20 <- approx.z(zt20, times20,times200) yp <- predict(fit1s20,newdata=pred.df) lines(times200,yp ,col=4) # mixed model spline terms at multiple levels of grouping data(spruce) Spruce$Zday <- smspline(~ days, data=spruce) Spruce$all <- rep(1,nrow(spruce)) # overall spline term, random plot and Tree effects spruce.fit1 <- lme(logsize ~ days, data=spruce, random=list(all= pdident(~zday -1), plot=~1, Tree=~1)) # try overall spline term plus plot level linear + spline term spruce.fit2 <- lme(logsize ~ days, data=spruce, random=list(all= pdident(~zday - 1), plot= pdblocked(list(~ days,pdident(~zday - 1))), Tree = ~1)) anova(spruce.fit1,spruce.fit2) summary(spruce.fit1) smsplineex1 Simulated data about a smooth curve Description Simulated data to demonstrate smoothing spline curve fitting with smspline and lme Usage data(smsplineex1) Format A data frame with 100 observations on the following 4 variables. time time covariate

6 6 smsplineex1 Details y simulated response values y.true true response values 100 data points were simulated about the curve y = 10 6 exp( 4t/100), with iid normal random errors with standard deviation 1. Examples data(smsplineex1)

7 Index Topic datasets smsplineex1, 5 Topic manip approx.z, 1 Topic models smspline, 3 Topic regression smspline, 3 Topic smooth smspline, 3 approx.z, 1, 4 lme, 5 nlme, 2, 4 predict.lme, 4 smspline, 2, 3, 5 smsplineex1, 5 7